jaroslav hajek and asymptotic theory of rank tests - Kybernetika

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246 J. JURECKOVA 6. FURTHER ASYMPTOTIC PROPERTIES OF RANKS Hajek [9] demonstrated that not only the best Pitman efficiency but also the best exact Bahadur slope is attainable by rank statistics; otherwise speaking, that the vector of ranks is sufficient in the Bahadur sense. Consider the two-sample model with independent samples X\,..., X n and Y\,... . .. ,Y m with the respective densities and d.f.'s /, g, F, G; let limAr^oo w+n = \ £ (0,1) and denote H(x) = \F(x) + (l-\)G(x), xeRi, (42) J(u)=-^-F(H- l (u)), g( u )=±G(H- 1 (u)), 0
Jaroslav Hájek and Asymptotic Theory of Rank Tests 247 7. EXTENSION OF RANK-SCORES PROCESS TO REGRESSION MODEL Hajek's results and methods were used and extended by a host of statisticians; it is impossible to characterize all this work as a whole. Among many possible extensions, let us briefly describe a recent extension of Hajek's rank-scores process to linear regression model, which in turn has further interesting applications. Consider the linear regression model Yi = x'i(3 + Ei, i=l,...,n (50) with Xi € R p , xn = 1, t = 1,... ,p and with independent errors E\,... ,E n . Koenker and Bassett [25] introduced the a-regression quantile (3(a) (0 < a < 1) for model (50) as a solution of the minimization where n ^p a (Yi - x'ib) •- min, 6
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